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Set coverings and invertibility of functional Galois connections. (English) Zbl 1080.06001

Litvinov, G. L. (ed.) et al., Idempotent mathematics and mathematical physics. Proceedings of the international workshop, Vienna, Austria, February 3–10, 2003. Providence, RI: American Mathematical Society (AMS) (ISBN 0-8218-3538-6/pbk). Contemporary Mathematics 377, 19-51 (2005).
Summary: We consider equations of the form \(Bf=g\), where \(B\) is a Galois connection between lattices of functions. This includes the case where \(B\) is the Legendre-Fenchel transform, or more generally a Moreau conjugacy. We characterise the existence and uniqueness of a solution \(f\) in terms of generalised subdifferentials. This extends a theorem of Vorobyev and Zimmermann relating solutions of max-plus linear equations and set coverings. We give various illustrations.
For the entire collection see [Zbl 1069.00011].

MSC:

06A15 Galois correspondences, closure operators (in relation to ordered sets)
49N15 Duality theory (optimization)
49J52 Nonsmooth analysis
44A15 Special integral transforms (Legendre, Hilbert, etc.)
46N10 Applications of functional analysis in optimization, convex analysis, mathematical programming, economics
90C25 Convex programming
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